mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: case left
μ : Measure ℝ
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : NullSingletonClass μ
b : ℝ
⊢ (fun x ↦ 0) =ᶠ[𝓝[<] 0] fun δ ↦ μ (Icc (b - δ) (b + δ))

Tactic:
filter_upwards [self_mem_nhdsWithin] with r (hr : r < 0)

Goal state after:
μ : Measure ℝ
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : NullSingletonClass μ
b r : ℝ
hr : r < 0
⊢ 0 = μ (Icc (b - r) (b + r))
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: X : Type u_3
Y : Type u_4
α : Type u_5
inst✝¹⁰ : Zero α
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : OpensMeasurableSpace X
inst✝⁴ : OpensMeasurableSpace Y
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetrizableSpace α
inst✝¹ : MeasurableSpace α
inst✝ : BorelSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, inst✝³] f
h'f : HasCompactSupport f
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
g : ℕ → SimpleFunc (X × Y) α
hg : ∀ (n : ℕ) (x : X × Y), dist (f x) ((g n) x) < u n
⊢ ∀ (x : X × Y), Tendsto (fun n ↦ (g n) x) atTop (𝓝 (f x))

Tactic:
intro x

Goal state after:
X : Type u_3
Y : Type u_4
α : Type u_5
inst✝¹⁰ : Zero α
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : OpensMeasurableSpace X
inst✝⁴ : OpensMeasurableSpace Y
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetrizableSpace α
inst✝¹ : MeasurableSpace α
inst✝ : BorelSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, inst✝³] f
h'f : HasCompactSupport f
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
g : ℕ → SimpleFunc (X × Y) α
hg : ∀ (n : ℕ) (x : X × Y), dist (f x) ((g n) x) < u n
x : X × Y
⊢ Tendsto (fun n ↦ (g n) x) atTop (𝓝 (f x))
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : TopologicalSpace β
m m₀ : MeasurableSpace α
μ : Measure α
f : α → β
m₀' : MeasurableSpace α
hm₀ : m₀' ≤ m₀
hf : AEStronglyMeasurable f (μ.trim hm₀)
⊢ AEStronglyMeasurable f μ

Tactic:
by
  obtain ⟨g, hg_meas, hfg⟩ := hf; exact ⟨g, hg_meas, ae_eq_of_ae_eq_trim hfg⟩

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x
r✝ : ℝ≥0
right✝ ha✝ : ↑r✝ < 1
ha : r✝ < 1
rs : α →ₛ ℝ≥0 := SimpleFunc.map (fun a ↦ r✝ * a) s
eq_rs : SimpleFunc.map c rs = const α ↑r✝ * SimpleFunc.map c s
eq : ∀ (p : ℝ≥0∞), ⇑(SimpleFunc.map c rs) ⁻¹' {p} = ⋃ n, ⇑(SimpleFunc.map c rs) ⁻¹' {p} ∩ {a | p ≤ f n a}
r : ℝ≥0∞
i j : ℕ
h : i ≤ j
⊢ ∀ (x : α), r ≤ f i x → r ≤ f j x

Tactic:
intro x

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x
r✝ : ℝ≥0
right✝ ha✝ : ↑r✝ < 1
ha : r✝ < 1
rs : α →ₛ ℝ≥0 := SimpleFunc.map (fun a ↦ r✝ * a) s
eq_rs : SimpleFunc.map c rs = const α ↑r✝ * SimpleFunc.map c s
eq : ∀ (p : ℝ≥0∞), ⇑(SimpleFunc.map c rs) ⁻¹' {p} = ⋃ n, ⇑(SimpleFunc.map c rs) ⁻¹' {p} ∩ {a | p ≤ f n a}
r : ℝ≥0∞
i j : ℕ
h : i ≤ j
x : α
⊢ r ≤ f i x → r ≤ f j x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
m₀ : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoMetrizableSpace β
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : IsEmbedding g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
this✝³ : MeasurableSpace β := borel β
this✝² : BorelSpace β
this✝¹ : MeasurableSpace γ := borel γ
this✝ : BorelSpace γ
H : AEStronglyMeasurable (fun x ↦ g (f x)) μ
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, f x ∈ t

Tactic:
rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 H).2 with ⟨t, ht, h't⟩
    exact ⟨g ⁻¹' t, hg.isSeparable_preimage ht, h't⟩

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
n : ℕ
x : α
hxt : x ∉ t
⊢ (fs n) x = 0

Tactic:
rw [SimpleFunc.restrict_apply _ ((hS_meas n).inter ht)]

Goal state after:
α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
n : ℕ
x : α
hxt : x ∉ t
⊢ (S n ∩ t).indicator (⇑(f_approx n)) x = 0
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
this : Monotone fun n ↦ ∫⁻ (x : α), f n x ∂μ
key : ∫⁻ (x : α), F x ∂μ = ⨆ n, ∫⁻ (x : α), f n x ∂μ
⊢ Tendsto (fun n ↦ ∫⁻ (x : α), f n x ∂μ) atTop (𝓝 (⨆ n, ∫⁻ (x : α), f n x ∂μ))

Tactic:
exact tendsto_atTop_iSup this

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
⊢ aeSeq hf p b₁ ≤ aeSeq hf p z

Tactic:
intro x

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
⊢ aeSeq hf p b₁ x ≤ aeSeq hf p z x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
⊢ aeSeq hf p b₂ ≤ aeSeq hf p z

Tactic:
intro x

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
⊢ aeSeq hf p b₂ x ≤ aeSeq hf p z x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
E : Type u_6
x✝ : MeasurableSpace α
inst✝⁵ : AddCancelMonoid E
inst✝⁴ : TopologicalSpace E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : ContinuousAdd E
inst✝ : PseudoMetrizableSpace E
g f : α → E
hg : Measurable g
φ : ℕ → α →ₛ E
hφ : ∀ (x : α), Tendsto (fun n ↦ (φ n) x) atTop (𝓝 (f x))
this : Tendsto (fun n x ↦ g x + (φ n) x) atTop (𝓝 (g + f))
⊢ Measurable (g + f)

Tactic:
apply measurable_of_tendsto_metrizable (fun n ↦ ?_) this

Goal state after:
α : Type u_5
E : Type u_6
x✝ : MeasurableSpace α
inst✝⁵ : AddCancelMonoid E
inst✝⁴ : TopologicalSpace E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : ContinuousAdd E
inst✝ : PseudoMetrizableSpace E
g f : α → E
hg : Measurable g
φ : ℕ → α →ₛ E
hφ : ∀ (x : α), Tendsto (fun n ↦ (φ n) x) atTop (𝓝 (f x))
this : Tendsto (fun n x ↦ g x + (φ n) x) atTop (𝓝 (g + f))
n : ℕ
⊢ Measurable fun x ↦ g x + (φ n) x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
E : Type u_6
x✝ : MeasurableSpace α
inst✝⁵ : AddCancelMonoid E
inst✝⁴ : TopologicalSpace E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : ContinuousAdd E
inst✝ : PseudoMetrizableSpace E
g f : α → E
hg : Measurable g
φ : ℕ → α →ₛ E
hφ : ∀ (x : α), Tendsto (fun n ↦ (φ n) x) atTop (𝓝 (f x))
this : Tendsto (fun n x ↦ (φ n) x + g x) atTop (𝓝 (f + g))
⊢ Measurable (f + g)

Tactic:
apply measurable_of_tendsto_metrizable (fun n ↦ ?_) this

Goal state after:
α : Type u_5
E : Type u_6
x✝ : MeasurableSpace α
inst✝⁵ : AddCancelMonoid E
inst✝⁴ : TopologicalSpace E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : ContinuousAdd E
inst✝ : PseudoMetrizableSpace E
g f : α → E
hg : Measurable g
φ : ℕ → α →ₛ E
hφ : ∀ (x : α), Tendsto (fun n ↦ (φ n) x) atTop (𝓝 (f x))
this : Tendsto (fun n x ↦ (φ n) x + g x) atTop (𝓝 (f + g))
n : ℕ
⊢ Measurable fun x ↦ (φ n) x + g x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
hf₀ : ¬∫⁻ (a : α), f a ∂μ = 0
⊢ ∃ g ≤ f, Measurable g ∧ μ (support g) < ∞ ∧ ∫⁻ (a : α), f a ∂μ - ε < ∫⁻ (a : α), g a ∂μ

Tactic:
obtain ⟨g, hgf, hgε⟩ : ∃ (g : α →ₛ ℝ≥0∞) (_ : g ≤ f), ∫⁻ a, f a ∂μ - ε < g.lintegral μ := by
      simpa only [← lt_iSup_iff, ← lintegral_def] using ENNReal.sub_lt_self hf hf₀ hε

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
hf₀ : ¬∫⁻ (a : α), f a ∂μ = 0
g : α →ₛ ℝ≥0∞
hgf : ⇑g ≤ f
hgε : ∫⁻ (a : α), f a ∂μ - ε < g.lintegral μ
⊢ ∃ g ≤ f, Measurable g ∧ μ (support g) < ∞ ∧ ∫⁻ (a : α), f a ∂μ - ε < ∫⁻ (a : α), g a ∂μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : One β
f : α → β
hf : Continuous[inst✝⁴, inst✝²] f
k : Set α
hk : IsCompact k
h'f : mulSupport f ⊆ k
⊢ StronglyMeasurable f

Tactic:
by
  borelize β
  let : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
  rw [stronglyMeasurable_iff_measurable_separable]
  exact ⟨hf.measurable, (isCompact_range_of_mulSupport_subset_isCompact hf hk h'f).isSeparable⟩

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
ι : Type u_2
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → ℝ≥0∞
f bound : α → ℝ≥0∞
hF_meas : ∀ᶠ (n : ι) in l, AEMeasurable (F n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a
h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ∞
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) l (𝓝 (f a))
⊢ ∀ (x : ℕ → ι), Tendsto x atTop l → Tendsto ((fun n ↦ ∫⁻ (a : α), F n a ∂μ) ∘ x) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))

Tactic:
intro x

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
ι : Type u_2
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → ℝ≥0∞
f bound : α → ℝ≥0∞
hF_meas : ∀ᶠ (n : ι) in l, AEMeasurable (F n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a
h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ∞
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) l (𝓝 (f a))
x : ℕ → ι
⊢ Tendsto x atTop l → Tendsto ((fun n ↦ ∫⁻ (a : α), F n a ∂μ) ∘ x) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
ι : Type u_2
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → ℝ≥0∞
f bound : α → ℝ≥0∞
hF_meas : ∀ᶠ (n : ι) in l, AEMeasurable (F n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a
h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ∞
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ (b : ℕ), a ≤ b → x b ∈ s
⊢ Tendsto ((fun n ↦ ∫⁻ (a : α), F n a ∂μ) ∘ x) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))

Tactic:
have h := inter_mem hF_meas h_bound

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
ι : Type u_2
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → ℝ≥0∞
f bound : α → ℝ≥0∞
hF_meas : ∀ᶠ (n : ι) in l, AEMeasurable (F n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a
h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ∞
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ (b : ℕ), a ≤ b → x b ∈ s
h : {x | (fun n ↦ AEMeasurable (F n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} ∈ l
⊢ Tendsto ((fun n ↦ ∫⁻ (a : α), F n a ∂μ) ∘ x) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : MeasurableSingletonClass α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
h : SecondCountableTopologyEither α β
f : α → β
hf : {x | ¬ContinuousAt f x}.Countable
⊢ StronglyMeasurable f

Tactic:
have : ContinuousOn f {x | ContinuousAt f x} := fun x hx ↦ hx.continuousWithinAt
  apply this.stronglyMeasurable_of_countable_compl
  convert hf
  grind

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β
g : α → γ
g' : γ → β
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝ : TopologicalSpace β
hg : MeasurableEmbedding g
hf : StronglyMeasurable f
hg' : StronglyMeasurable g'
⊢ ∀ (x : γ),
    Tendsto (fun n ↦ ((fun n ↦ (hf.approx n).extend g hg (hg'.approx n)) n) x) atTop (𝓝 (Function.extend g f g' x))

Tactic:
intro x

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β
g : α → γ
g' : γ → β
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝ : TopologicalSpace β
hg : MeasurableEmbedding g
hf : StronglyMeasurable f
hg' : StronglyMeasurable g'
x : γ
⊢ Tendsto (fun n ↦ ((fun n ↦ (hf.approx n).extend g hg (hg'.approx n)) n) x) atTop (𝓝 (Function.extend g f g' x))
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
s : Set α
f : α → β
hs : MeasurableSet s
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
⊢ ∀ (x : α), Tendsto (fun n ↦ ((fun n ↦ (hf.approx n).restrict s) n) x) atTop (𝓝 (f x))

Tactic:
intro x

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
s : Set α
f : α → β
hs : MeasurableSet s
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
x : α
⊢ Tendsto (fun n ↦ ((fun n ↦ (hf.approx n).restrict s) n) x) atTop (𝓝 (f x))
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
s : Set α
f : α → β
hs : MeasurableSet s
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
⊢ ∀ x ∉ s, ∀ (n : ℕ), ((fun n ↦ (hf.approx n).restrict s) n) x = 0

Tactic:
intro x

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
s : Set α
f : α → β
hs : MeasurableSet s
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
x : α
⊢ x ∉ s → ∀ (n : ℕ), ((fun n ↦ (hf.approx n).restrict s) n) x = 0
Module: Mathlib.Topology.Perfect

Goal state before:
pp: case acc
α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
hC : Preperfect C
⊢ Preperfect (closure[inst✝] C)

Tactic:
intro x

Goal state after:
case acc
α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
hC : Preperfect C
x : α
⊢ x ∈ closure[inst✝] C → AccPt x (𝓟 (closure[inst✝] C))
Module: Mathlib.Topology.Perfect

Goal state before:
pp: case mpr
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : T1Space α
h : Perfect (closure[inst✝¹] C)
⊢ Preperfect C

Tactic:
intro x

Goal state after:
case mpr
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : T1Space α
h : Perfect (closure[inst✝¹] C)
x : α
⊢ x ∈ C → AccPt x (𝓟 C)
Module: Mathlib.Topology.Perfect

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T1Space α
U : Set α
hu : U.Nontrivial
h : IsPreconnected U
⊢ Preperfect U

Tactic:
intro x

Goal state after:
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T1Space α
U : Set α
hu : U.Nontrivial
h : IsPreconnected U
x : α
⊢ x ∈ U → AccPt x (𝓟 U)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), Measurable (u i)
⊢ ∃ t, ∀ (j : ι) (x : α), Tendsto (fun n ↦ u ((t n) j) x) atTop (𝓝 (u j x))

Tactic:
have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    refine ⟨h_str_meas.approx, fun j x => ?_⟩
    exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)

Goal state after:
Module: Mathlib.Topology.Perfect

Goal state before:
pp: case refine_2
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed[inst✝¹] C
b : Set (Set α)
bct : b.Countable
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ (U ∩ C).Countable}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : (V ∩ C).Countable
⊢ ∀ x ∈ D, ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ D, y ≠ x

Tactic:
intro x

Goal state after:
case refine_2
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed[inst✝¹] C
b : Set (Set α)
bct : b.Countable
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ (U ∩ C).Countable}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : (V ∩ C).Countable
x : α
⊢ x ∈ D → ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ D, y ≠ x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t ↦ Nat.rec (motive := fun x ↦ α → ℝ≥0∞) (g 0) (fun n I x ↦ max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
⊢ ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x

Tactic:
filter_upwards [hf_mono] with x hx n

Goal state after:
α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t ↦ Nat.rec (motive := fun x ↦ α → ℝ≥0∞) (g 0) (fun n I x ↦ max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
x : α
hx : Monotone fun i ↦ f i x
n : ℕ
⊢ g' n x ≤ f n x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ ∃ t, ∀ (j : ι) (x : α), Tendsto (fun n ↦ u ((t n) j) x) atTop (𝓝 (u j x))

Tactic:
have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    refine ⟨h_str_meas.approx, fun j x => ?_⟩
    exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: case h
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
s : Set α
hs : IsCompact s
hf' : ContinuousOn f s
ε : (x : α) → x ∈ s → ℝ
hε : ∀ (x : α) (a : x ∈ s), ε x a > 0
K : (x : α) → x ∈ s → ℝ≥0
hf : ∀ (x : α) (a : x ∈ s), LipschitzOnWith (K x a) f (ball x (ε x a) ∩ s)
K' : (x : α) → x ∈ s → ℝ≥0
hK' : ∀ (x : α) (hx : x ∈ s), ∀ y ∈ s.diff (ball x (ε x hx)), edist (f x) (f y) ≤ ↑(K' x hx) * edist x y
t : Finset ↑s
ht : s ⊆ ⋃ x ∈ t, s ∩ ball (↑x) (ε ↑x ⋯ / 2)
⊢ LipschitzOnWith (t.sup fun i ↦ K ↑i ⋯ + 2 * K' ↑i ⋯) f s

Tactic:
intro x

Goal state after:
case h
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
s : Set α
hs : IsCompact s
hf' : ContinuousOn f s
ε : (x : α) → x ∈ s → ℝ
hε : ∀ (x : α) (a : x ∈ s), ε x a > 0
K : (x : α) → x ∈ s → ℝ≥0
hf : ∀ (x : α) (a : x ∈ s), LipschitzOnWith (K x a) f (ball x (ε x a) ∩ s)
K' : (x : α) → x ∈ s → ℝ≥0
hK' : ∀ (x : α) (hx : x ∈ s), ∀ y ∈ s.diff (ball x (ε x hx)), edist (f x) (f y) ≤ ↑(K' x hx) * edist x y
t : Finset ↑s
ht : s ⊆ ⋃ x ∈ t, s ∩ ball (↑x) (ε ↑x ⋯ / 2)
x : α
⊢ x ∈ s → ∀ ⦃y : α⦄, y ∈ s → edist (f x) (f y) ≤ ↑(t.sup fun i ↦ K ↑i ⋯ + 2 * K' ↑i ⋯) * edist x y
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: case inl
x : ℝ
h₀ : x ≤ 0
⊢ ↑(σ (projIcc 0 1 ⋯ x)) = ↑(projIcc 0 1 ⋯ (1 - x))

Tactic:
simp [projIcc_of_le_left, projIcc_of_right_le, h₀]

Goal state after:
Module: Mathlib.Topology.Metrizable.CompletelyMetrizable

Goal state before:
pp: X✝ : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : Countable ι
X : ι → Type u_4
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), IsCompletelyPseudoMetrizableSpace (X i)
⊢ IsCompletelyPseudoMetrizableSpace ((i : ι) → X i)

Tactic:
let := fun i ↦ upgradeIsCompletelyPseudoMetrizable (X i)
  infer_instance

Goal state after:
Module: Mathlib.Topology.Metrizable.CompletelyMetrizable

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun x y ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
⊢ CompleteSpace X

Tactic:
· refine Metric.complete_of_cauchySeq_tendsto fun u hu ↦ ?_
    rw [Metric.cauchySeq_iff'] at hu
    obtain ⟨N, hN⟩ := hu 1 (by simp)
    refine ⟨u N, @tendsto_atTop_of_eventually_const X UniformSpace.toTopologicalSpace (u N) _ _ _ N
      fun n hn ↦ ?_⟩
    specialize hN n hn
    by_contra
    change (if u n = u N then 0 else 1) < 1 at hN
    simp_all

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: α : Type u_2
⊢ Injective res

Tactic:
intro x

Goal state after:
α : Type u_2
x : ℕ → α
⊢ ∀ ⦃a₂ : ℕ → α⦄, res x = res a₂ → x = a₂